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Ifm formula sheet
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coachingactuaries Copyright © 2019 Coaching Actuaries. All Rights Reserved. 1
Exam IFM
updated 12/20/
coachingactuaries Copyright © 2019 Coaching Actuaries. All Rights Reserved. 1
INTRODUCTION TO DERIVATIVES
Reasons for Using Derivatives ï To manage risk ï To speculate ï To reduce transaction cost ï To minimize taxes / avoid regulatory issues
Bid-ask Spread ï Bid price: The price at which market- makers will buy and end-users will sell. ï Ask/Offer price: The price at which market-makers will sell and end-users will buy. ï Bid-ask spread = Ask price – Bid price ï Round-trip transaction cost: Difference between what you pay and what you receive from a sale using the same set of bid/ask prices.
Payoff and Profit ï Payoff: Amount that one party would have if completely cashed out. ï Profit: Accumulated value of cash flows at the risk-free rate.
Long vs. Short ï A long position in an asset benefits from an increase in the price of the asset. ï A short position in an asset benefits from a decrease in the price of the asset.
Short-Selling Process of short-selling: ï Borrow an asset from a lender ï Immediately sell the borrowed asset and receive the proceeds (usually kept by lender or a designated 3rd party) ï Buy the asset at a later date in the open market to repay the lender (close/cover the short position)
Haircut: Additional collateral placed with lender by short-seller. It belongs to the short-seller.
Interest rate on haircut is called: ï short rebate in the stock market ï repo rate in the bond market
Reasons for short-selling assets: ï Speculation – To speculate that the price of a particular asset will decline. ï Financing – To borrow money for additional financing of a corporation. ï Hedging – To hedge the risk of a long position on the asset.
Option Moneyness ï In-the-money: Produce a positive payoff (not necessarily positive profit) if the option is exercised immediately. ï At-the-money: The spot price is approximately equal to the strike price. ï Out-of-the-money: Produce a negative payoff if the option is exercised immediately.
Option Exercise Styles ï European-style options can only be exercised at expiration. ï American-style options can be exercised at any time during the life of the option. ï Bermudan-style options can be exercised during bounded periods (i., specified periods during the life of the option).
Zero-coupon Bond (ZCB) Buying a risk-free ZCB = Lending at risk-free rate Selling a risk-free ZCB = Borrowing at risk-free rate Payoff on a risk-free ZCB = ZCB’s maturity value Profit on a risk-free ZCB = 0
INTRODUCTION TO DERIVATIVES
FORWARD CONTRACTS, CALL OPTIONS, AND PUT OPTIONS
Contract
Position in Contract
Description
Position in Underlying Payoff Profit
Maximum Loss
Maximum Gain Strategy
Forward
Long Forward
Obligation to buy at the forward price
Long S" − F%," Payoff F%," ∞
Guarantee/lock in purchase price of underlying
Short Forward
Obligation to sell at the forward price
Short F%," − S" Payoff ∞ F%,"
Guarantee/lock in sale price of underlying
Call
Long Call
Right (but not obligation) to buy at the strike price
Long max [0, S" − K] Payoff − AV(Prem. )
AV(Prem. ) ∞
Insurance against high underlying price
Short Call
Obligation to sell at the strike price if the call is exercised
Short −max [0, S" − K] Payoff + AV(Prem. )
∞ AV(Prem. )
Sells insurance against high underlying price
Put
Long Put
Right (but not obligation) to sell at the strike price
Short max [0, K − S"]
Payoff − AV(Prem. ) AV(Prem. )
K
−AV(Prem. )
Insurance against low underlying price
Short Put
Obligation to buy at the strike price if the put is exercised
Long −max [0, K − S"] Payoff + AV(Prem. )
K
−AV(Prem. )
AV(Prem. )
Sells insurance against low underlying price
FORWARD CONTRACTS, CALL OPTIONS, AND PUT OPTIONS
FORWARDS
4 Ways to Buy a Share of Stock
Ways
Payment Time at TimeStockReceive
Payment
Outright purchase 0 0 S% Fully leveraged purchase
T 0 S%eA"
Prepaid forward contract
0 T F%,"B (S)
Forward contract
T T F%,"(S)
Relationship between 𝐅𝐅𝐭𝐭,𝐓𝐓(𝐒𝐒) and 𝐅𝐅𝐭𝐭,𝐓𝐓𝐏𝐏 (𝐒𝐒)
FH,"(S) = Accumulated Value of FH,"B (S) = FH,"B (S) ⋅ eA("OH)
Dividend Structure 𝐅𝐅𝐭𝐭,𝐓𝐓𝐏𝐏 (𝐒𝐒) No Divs SH Discrete Divs SH − PVH,"(Divs) Continuous Divs SHeOT("OH)
Dividend Structure 𝐅𝐅𝐭𝐭,𝐓𝐓(𝐒𝐒) No Divs SHeA("OH)
Discrete Divs SHe
A("OH) − AVH,"(Divs)
Continuous Divs SHe(AOT)("OH)
Forward premium = F S%," % Annualized forward premium rate
= 1T ln F S%," %
Synthetic Forward Synthetic long forward is created by: ï buying a stock and borrowing money (i., selling a bond), or ï buying a call and selling a put at the same strike.
Synthetic short forward is the opposite, created by: ï selling a stock and lending money (i., buying a bond), or ï selling a call and buying a put at the same strike.
Arbitrage A transaction which generates a positive cash flow either today or in the future by simultaneous buying and selling of related assets, with no net investment or risk. Arbitrage strategy: “Buy Low, Sell High.”
Cash-and-Carry The actual forward is overpriced. Short actual forward + Long synthetic forward
Reverse Cash-and-Carry The actual forward is underpriced. Long actual forward + Short synthetic forward
FUTURES
Futures Compared to Forward ï Traded on an exchange ï Standardized (size, expiration, underlying) ï More liquid ï Marked-to-market and often settled daily ï Minimal credit risk ï Price limit is applicable
Features of Futures Contract Notional Value = # Contracts × Multipler × Futures price Balt = BaltO_ ⋅ eA` + GainH where ï GainH = # Contracts × Multipler × Price Changet (for 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 position) ï GainH = − # Contracts × Multipler × Price Changet (for 𝑠𝑠ℎ𝑙𝑙𝑜𝑜𝑜𝑜 position) ï Price Changet = Future PriceH − Future PriceHO_
Margin Call ï Maintenance margin: Minimum margin balance that the investor is required to maintain in margin account at all times ï Margin call: If the margin balance falls below the maintenance margin, then the investor will get a request for an additional margin deposit. The investor has to add more funds to bring the margin balance back to the initial margin.
PUT-CALL PARITY (PCP)
PCP for Stocks C(S, K) − P(S, K) = FH,"B (S) − KeOA("OH)
PCP for Futures C(F, K) − P(F, K) = FeOA("OH) − KeOA("OH)
PCP for Bonds C(B, K) − P(B, K) = FH,"B (B) − KeOA("OH) where FH,"B (B) = BH − PVH,"(Coupons) BH = Bond price at time t
PCP for Exchange Options 𝐂𝐂(𝐀𝐀, 𝐁𝐁) 𝐏𝐏(𝐀𝐀, 𝐁𝐁) receive A, give up B give up A, receive B
C(A, B) − P(A, B) = FH,"B (A) − FH,"B (B)
C(A, B) = P(B, A)
PCP for Currency Options Use the generalized PCP for exchange option.
For example, the prepaid forward price for 1 yen denominated in dollars is:
$F%,"B (¥1) = qx%
$
¥r (¥1e
OA¥") = $x%eOA¥"
Alternatively: S% → x% r → rt δ → rv Ct(f, K) − Pt(f, K) = x%eOAw" − KeOAx" where x% is in d/f
FORWARDS
FUTURES
PUT-CALL PARITY (PCP)
COMPARING OPTIONS
Bounds of Option Prices Call and Put S ≥ C{|}A ≥ C~A ≥ max(0, FB(S) − KeOA")
K ≥ P{|}A ≥ P~A ≥ max Ä0, KeOA" − FB(S)Å
European vs. American Call FB(S) ≥ C~A ≥ max(0, FB(S) − KeOA") S ≥ C{|}A ≥ max (0, S − K)
European vs. American Put
KeOA" ≥ P~A ≥ max Ä0, KeOA" − FB(S)Å
K ≥ P{|}A ≥ max (0, K − S)
Early Exercise of American Option PV(Interest on strike) = K(1 − eOA") PV(Divs) = SÑ1 − eOT"Ö
American Call ï Nondividend-paying stock o Early exercise is never optimal. o C{|}A = C~A ï Dividend-paying stock o It is rational to early exercise if: PV(Divs) > PV(Interest on strike) + Implicit Put o It may be rational to early exercise if: PV(Divs) > PV(Interest on strike)
American Put It is rational to early exercise if: PV(Interest on strike) > PV(Divs) + Implicit Call It may be rational to early exercise if: PV(Interest on strike) > PV(Divs)
Strike Price Effects For K_ < Kà < Kâ:
Call C(K_) ≥ C(Kà) ≥ C(Kâ) C(K_) − C(Kà) ≤ Kà − K_ European: C(K_) − C(Kà) ≤ PV(Kà − K_) C(K_) − C(Kà) Kà − K_ ≥ C(K
à) − C(Kâ) Kâ − Kà
Put P(K_) ≤ P(Kà) ≤ P(Kâ) P(Kà) − P(K_) ≤ Kà − K_ European: P(Kà) − P(K_) ≤ PV(Kà − K_) P(Kà) − P(K_) Kà − K_ ≤ P(K
â) − P(Kà) Kâ − Kà
Time Until Expiration For T_ < Tà: C{|}A(S, K, T_) ≤ C{|}A(S, K, Tà) P{|}A(S, K, T_) ≤ P{|}A(S, K, Tà)
For a nondividend-paying stock:
CA(S, K, T_) ≤ CA(S, K, Tà)
This is also generally true for European call options on dividend-paying stocks and European puts, with some exceptions.
BINOMIAL MODEL
Option Pricing: Replicating Portfolio An option can be replicated by buying Δ shares of the underlying stock and lending 𝐵𝐵 at the risk-free rate. Δ = eOT` ç V
− Vt S(u − d)é B = e
OA` çuVt − dV u − d é V = ΔS + B
To replicate a call, buy shares and borrow money. To replicate a put, sell shares and lend money.
Option Pricing: Risk-neutral Valuation
p∗ = e
(AOT) − d u − d V% = eOA ⋅ E∗[Payoff]
= eOA`[(p∗)V
- (1 − p∗)Vt] S%e(AOT)` = (p∗)S + (1 − p∗)St
Constructing a Binomial Tree General Method u = S S %
d = S St
%
Standard Binomial Tree
This is the usual method in McDonald based
on forward prices.
u = e(AOT)ëí√ d = e(AOT)Oí√
p∗ =
1
1 + eí√` Probability For n periods, let k be the number of "up" jumps needed to reach an ending node. Then, the risk-neutral probability of reaching that node is given by: ÄnkÅ (p∗)î(1 − p∗)ïOî, k = 0,1, ... , n
No-Arbitrage Condition Arbitrage is possible if the following inequality is not satisfied: 0 < p∗ < 1 ⟺ d < e(AOT)` < u
Option on Currencies
Substitutions: S% → x% r → rt δ → rv
u = e(AxOAw)ëí√ d = e(AxOAw)Oí√
p∗ = e
(AxOAw)` − d u − d
Option on Futures Contracts FH,"ö = SHe(AOT)("öOH) T = Expiration date of the option Tú = Expiration date of the futures contract T ≤ Tú
Substitutions: SH → FH,"ö δ → r
uú = eí√ dú = eOí√
p∗ = 1 − d u ú
ú − dú
Δ =
V − Vt F(uú − dú) B = eOA`[p∗V + (1 − p∗)Vt]
Call Put Δ + − B − +
COMPARING OPTIONS
BINOMIAL MODEL
Generalized BS Formula Assume the current time is time 0 and the options expire at time T: C = FB(S) ⋅ N(d_) − FB(K) ⋅ N(dà) P = FB(K) ⋅ N(−dà) − FB(S) ⋅ N(−d_)
d_ =
ln çF
B(S)
FB(K)é + 1 2 σ
àT
σ√T
dà =
ln çF
B(S)
FB(K)é − 1 2 σ
àT
σ√T
= d_ − σ√T
σ = √Var{ln[SH
]}
t , 0 < t ≤ T
= √Var∆ln•FH,"
(S)¶«
t , 0 < t ≤ T
= √Var∆ln•FH,"
B (S)¶«
t , 0 < t ≤ T
The generalized BS formula can be applied to various assets, including stocks, futures contracts, and currencies.
If the stock pays discrete dividends, then the volatility of the prepaid forward price should be used as the volatility parameter.
For a stock that pays continuous dividends, the generalized BS formula can be written as: C = S%eOT" ⋅ N(d_) − KeOA" ⋅ N(dà) P = KeOA" ⋅ N(−dà) − S%eOT" ⋅ N(−d_)
d_ = ln ÄS
% K Å + Är − δ + 1 2 σ
àÅ T σ√T
dà = ln ÄS
% K Å + Är − δ − 1 2 σ
àÅ T
σ√T = d_ − σ√T
Options on Futures Contract Use the generalized BS formula in conjunction with the appropriate prepaid forward prices.
The prepaid forward price for a futures contract is just the present value of the futures price:
F%,"B (F) = FeOA"
Alternatively: S% → F%,"ö δ → r C = F%,"ö eOA" ⋅ N(d_) − KeOA" ⋅ N(dà) P = KeOA" ⋅ N(−dà) − F%,"ö eOA" ⋅ N(−d_)
d_ =
ln çF%," K öé + 1 2 σàT σ√T
dà =
ln çF%," K öé − 1 2 σàT σ√T
= d_ − σ√T
Options on Currencies Use the generalized BS formula in conjunction with the appropriate prepaid forward prices.
For example, the prepaid forward price for 1 yen denominated in dollars is:
$F%,"B (¥1) = qx%
$
¥r (¥1e
OA¥") = $x%eOA¥"
Alternatively: S% → x% r → rt δ → rv C = x%eOAw" ⋅ N(d_) − KeOAx" ⋅ N(dà) P = KeOAx" ⋅ N(−dà) − x%eOAw" ⋅ N(−d_)
d_ = ln Äx
% K Å + Ärt − rv + 1 2 σ
àÅ T
σ√T
dà = ln Äx
% K Å + Ärt − rv − 1 2 σ
àÅ T
σ√T = d_ − σ√T
OPTION GREEKS
Greek Definition
Long Call
Long Put
Δ ∂V ∂S
+ −
Γ ∂Δ
∂S = ∂
àV ∂Sà
+ +
θ ∂V∂t − * −*
Vega ∂V ∂σ
+ +
ρ ∂V ∂r
+ −
ψ ∂V ∂δ
− +
- θ is usually negative. Note: For short positions, just reverse the signs.
Option Greeks Formulas The formulas for the six option Greeks for both call and put options under the BS framework as well as NÕ(x) will be provided on the exam.
ΔŒ = eOT("OH)N(d_), 0 ≤ ΔŒ ≤ 1 ΔB = −eOT("OH)N(−d_), −1 ≤ ΔB ≤ 0 ΔŒ − ΔB = eOT("OH)
ΓŒ = ΓB = expÑ−δ(T − t)ÖN
Õ(d_) Sσ√T − t θŒ = δSeOT("OH)N(d_) − rKeOA("OH)N(dà)
− Ke
OA("OH)NÕ(dà)σ 2√T − t θB = θŒ + rKeOA("OH) − δSeOT("OH) VegaŒ = VegaB = SeOT("OH)NÕ(d_)√T − t ρŒ = (T − t)KeOA("OH)N(dà) ρB = −(T − t)KeOA("OH)N(−dà) ψŒ = −(T − t)SeOT("OH)N(d_) ψB = (T − t)SeOT("OH)N(−d_)
NÕ(x) = e
Oœ§/à √2π
Risk Premium The risk premium of an asset is defined as the excess of the expected return of the asset over the risk-free return: ï Risk Premium“”Hæ‘ï = γ − r ï Risk Premium≠H‘’î = α − r
γ − r = Ω(α − r) σ“”Hæ‘ï = |Ω|σ≠H‘’î
Sharpe Ratio φ“”Hæ‘ï = Option
Õs risk premium OptionÕs volatility =
γ − r σ“”Hæ‘ï
φ≠H‘’î = Stock
Õs risk premium StockÕs volatility = α − rσ≠H‘’î φŒ = φ≠H‘’î; φB = −φ≠H‘’î
OPTION GREEKS
Elasticity
Ω = % change in option price% change in stock price = Δ ⋅ SV
ΩŒ ≥ 1; ΩB ≤ 0
Portfolio Greek & Elasticity Greek for a portfolio = sum of the Greeks Elasticity for a portfolio = weighted average of the elasticities
ΩB‘AH = ΔB‘AH
⋅ S
VB‘AH = ⁄ ωæΩæ
ï
æ¡_ γB‘AH − r = ΩB‘AH(α − r)
Delta-Gamma-Theta Approximation
VHë ≈ VH + ΔHε + 1 2 ΓHεà + θHh; ε = SHë − SH
DELTA & GAMMA HEDGING
Overnight Profit A delta-hedged portfolio has 3 components: ï Buy/sell options ï Buy/sell stocks ï Borrow/lend money (sell/buy bond)
Overnight profit is the sum of: ï Profit on options bought/sold ï Profit on stocks bought/sold ï Profit on bond
Alternatively, overnight profit is the sum of: ï Gain on options, ignoring interest ï Gain on stocks, ignoring interest ï Interest on borrowed/lent money
For a market-maker who writes an option and delta-hedges the position, the market- maker’s profit from time t to t + h is:
= ΔH(SHë − SH) − (VHë − VH)
− ÑeA` − 1Ö(ΔHSH − VH)
≈ − 1 2 εàΓH − hθH − ÑeA` − 1Ö(ΔHSH − VH)
where ε = SHë` − SH
If h is small, then eA` − 1 ≈ rh.
Breakeven If the price of the underlying stock changes by one standard deviation over a short period of time, then a delta-hedged portfolio does not produce profits or losses.
Assuming the BS framework, given the current stock price, S, the two stock prices after a period of h for which the market- maker would break even are: S ± Sσ√h
Multiple Greeks Hedging Δ≠H‘’î = 1; all other Greeks of the stock = 0 To hedge multiple Greeks, set the sum of the Greeks you are hedging to zero.
ACTUARIAL-SPECIFIC RISK
MANAGEMENT
Options Embedded in Insurance Products ï A guaranteed minimum death benefit (GMDB) guarantees a minimum amount will be paid to a beneficiary when the policyholder dies. ï A guaranteed minimum accumulation benefit (GMAB) guarantees a minimum value for the underlying account after some period of time, even if the account value is less. ï A guaranteed minimum withdrawal benefit (GMWB) guarantees that upon the policyholder reaching a certain age, a minimum withdrawal amount over a specified period will be provided. ï A guaranteed minimum income benefit (GMIB) guarantees the purchase price of a traditional annuity at a future time.
GMDB with a Return of Premium Guarantee ï A guarantee which returns the greater of the account value and the original amount invested: max(S", K) = S" + max( K − S", 0) ï The embedded option is a put option. Its value is: E[P(Tœ)] = fl P(t)f"‡ (t)dt
· %
Earnings-Enhanced Death Benefit ï Pays the beneficiary an amount based on the increase in the account value over the original amount invested, e., 40% ⋅ max( S" − K, 0). ï The embedded option is a call option. Its value is: E[C(Tœ)] = fl C(t)f"‡ (t)dt
· %
GMAB with a Return of Premium Guarantee ï Similar to GMDB with ROP guarantee, but the benefit is contingent on the policyholder surviving to the end of the guarantee period. ï The embedded option is a put option. Its value is: P(m) ⋅ Pr[T†∗ ≥ m]
Mortgage Loan as Put For an uninsured position, the loss to the mortgage lender is max(B + C∗ − R, 0), where: ï B is the outstanding loan balance at default ï C∗is the lender’s total settlement cost ï R is the amount recovered on the sale of property This is a put payoff with K = B + C∗ and S = R.
Static vs. Dynamic Hedging Static/hedge-and-forget: Buy options and hold to expiration Dynamic: Frequently buy/sell assets and/or derivatives with the goal of matching changes in the value of guarantee
Hedging of Catastrophic Risk Catastrophe bond: A bond issued to investors where repayments and principal payments are contingent on there not being a catastrophe which causes large losses for the insurer. Thus, investors who buy these bonds face the risk of not receiving coupon payments or repayment of their principal. In general, cat bondholders typically receive higher interest rates for taking on this risk.
DELTA & GAMMA HEDGING ACTUARIAL-SPECIFIC
RISK MANAGEMENT
Chooser Option For an option that allows the owner to choose at time t whether the option will become a European call or put with strike K and expiring at time T:
VH = max[C(SH, K, T − t), P(SH, K, T − t)] = eOT("OH) max•0, KeO(AOT)("OH) − SH¶ + C(SH, K, T − t) The time-0 value is:
V% = eOT("OH) ⋅ PÑS%, KeO(AOT)("OH), tÖ + C(S%, K, T)
Lookback Option An option whose payoff at expiration depends on the maximum or minimum of the stock price over the life of the option.
Type Payoff Standard lookback call S" − min(S) Standard lookback put
max(S) − S"
Extrema lookback call max[0, max(S) − K] Extrema lookback put max[0, K − min(S)]
Standard lookback options are known as lookback options with a floating strike price.
Extrema lookback options are known as lookback options with a fixed strike price.
Shout Option An option that gives the owner the right to lock in a minimum payoff exactly once during the life of the option, at a time that the owner chooses. When the owner exercises the right to lock in a minimum payoff, the owner is said to shout to the writer.
S∗ is the value of the stock at the time when the option owner shouts to the option writer.
Payoff for a shout call
= „max[S" − K, S
∗ − K, 0] if exercised max[S" − K, 0] if not exercised
Payoff for a shout put
= „max[K − S", K − S
∗, 0] if exercised max[K − S", 0] if not exercised
Rainbow Option An option whose payoff depends on two or more risky assets.
Hedging Strategies Using Exotic Options
- Forward start options are useful for hedging guarantees that will come into effect during the payout period of a GMWB while the variable annuity is still in the accumulation period.
- Chooser options are useful hedging tools for variable annuities with two-sided guarantees, e., a GMDB with a return-of- premium guarantee and an earnings- enhanced death benefit equal to 35% of any account value gains.
- Lookback options are useful for hedging variable annuity guarantees where the guarantee value is periodically recalculated as the greater of the account value and the existing guarantee value.
- Shout options are useful for hedging variable annuity guarantees in situations where the guarantee value is recalculated at the discretion of the policyholder.
- Rainbow options are useful hedging tools when policyholders can hold multiple assets in their accounts and the guarantee applies to the account as a whole rather than individual assets in the account.
MEAN-VARIANCE
PORTFOLIO THEORY
For Corporate Finance questions, unless you are told otherwise, assume that interest rates are annual effective, consistent with the Berk/DeMarzo text.
Risk and Return of a Single Asset
E[R] = ⁄ pæ ⋅ Ræ
ï
¡_ Var[R] = E[(R − E[R])à]
= ⁄ pæ ⋅ (Ræ − E[R])à
ï
¡_ = E[Rà] − (E[R])à SD[R] = ÒVar[R]
Realized Returns RHë_ = Capital Gain + Div Yield = PHë_
− PH
PH + D
Hë_ PH
Annual Realized Returns R = Ñ1 + RÚ_ÖÑ1 + RÚàÖÑ1 + RÚâÖÑ + RÚÛÖ − 1
Average Annual Returns Based on arithmetic average:
RÙ = 1T ⁄ RH
"
H¡_ Is used to estimate a stock’s expected return over a future horizon based on its past performance.
Compound Annual Returns Based on geometric average: RÙ = [(1 + R_)(1 + Rà) ... (1 + R")]
_" − 1 Is a better description of the long-run historical performance of a stock.
Variance of Returns
Var[R] = T − 1 1 ⁄(RH − RÙ)à
"
H¡_ Standard Error =
SD(R)
√# Observations 95% confidence interval for expected return = Average return ± 2 ⋅ Standard Error = RÙ ± 2 ⋅
SD(R)
√# Observations
Risk and Return of a Portfolio RB = x_R_ + xàRà + ⋯ + xïRï E[RB] = x_E[R_] + xàE[Rà] + ⋯ + xïE[Rï] xæ =
Value of Investment i Total Portfolio Value ; ∑xæ = 1 Cov•Ræ, R ̃¶ = E•(Ræ − E[Ræ])ÑR ̃ − E•R ̃¶Ö¶
= T − 1 1 ⁄ÑRæ,H − RÙæÖÑR ̃,H − RÙ ̃Ö
"
æ¡_ = ρæ, ̃ ⋅ σæ ⋅ σ ̃
For a 2-stock portfolio: RB = x_R_ + xàRà σBà = x_àσ_à + xààσàà + 2x_xàCov[R_, Rà]
For a 3-stock portfolio: RB = x_R_ + xàRà + xâRâ σBà = x_àσ_à + xààσàà + xâàσâà + 2x_xàCov[R_, Rà] + 2x_xâCov[R_, Râ] + 2xàxâCov[Rà, Râ]
MEAN-VARIANCE
PORTFOLIO THEORY
For an n-stock portfolio:
σBà = ⁄ xæCov[Ræ, RB]
ï
æ¡_
= ⁄ ⁄ xæx ̃Cov•Ræ, R ̃¶
ï
̃¡_
ï
æ¡_ In the covariance matrix, we have: ï n × n = nà total elements ï n variance terms ï nà − n true covariance terms ï (nà − n)/2 unique true covariance terms
Note that: ï Cov•Ræ, R ̃¶ = Cov•R ̃, Ræ¶ = ρæ, ̃ ⋅ σæ ⋅ σ ̃
ï Cov[Ræ, Ræ] = Var[Ræ] ï Cov[aR_ + bRà, cR_ + dRà] = acCov[R_, R_] + adCov[R_, Rà] +bcCov[Rà, R_] + bdCov[Rà, Rà] = acVar[R_] + adCov[R_, Rà] +bcCov[Rà, R_] + bdVar[Rà]
Diversification
Systematic risk ï Also known as common, market, or non-diversifiable risk. ï Fluctuations in a stock's return that are due to market-wide news.
Nonsystematic risk ï Also known as firm-specific, independent, idiosyncratic, unique, or diversifiable risk. ï Fluctuations in a stock's return that are due to firm-specific news.
Total risk = Systematic risk + Unsystematic risk
Diversification reduces a portfolio's total risk by averaging out nonsystematic fluctuations: ï Investors can eliminate nonsystematic risk for free by diversifying their portfolios. Thus, the risk premium for nonsystematic risk is zero. ï The risk premium of a security is determined by its systematic risk and does not depend on its nonsystematic risk.
For an equally-weighted n-stock portfolio: σBà = 1n ⋅ Var‚‚‚‚‚ + ç1 − 1né ⋅ Cov‚‚‚‚‚ ï In a very large portfolio (n → ∞), the covariance among the stocks accounts for the bulk of portfolio risk: σBà = Cov‚‚‚‚‚ ï If the stocks are independent and have identical risks, then Cov‚‚‚‚‚ = 0, and: σBà = 1n ⋅ Var‚‚‚‚‚ As n → ∞, σBà → 0. Thus, a very large portfolio with independent and identical risks will have zero risk.
Observations: ï The diversification effect is most significant initially. ï Even with a very large portfolio, we cannot eliminate all risk. The remaining risk is systematic risk that cannot be avoided through diversification.
For a portfolio with n individual stocks with arbitrary weights:
σB = ⁄ xæ ⋅ σæ ⋅ ρæ,B
ï
æ¡_ ï Each security contributes to the portfolio volatility according to its total risk scaled by its correlation with the portfolio, which adjusts for the fraction of the total risk that is common to the portfolio. ï As long as the correlation is not +1, the volatility of the portfolio is always less than the weighted average volatility of the individual stocks.
Mean-Variance Portfolio Theory
Assumptions of Mean-Variance Analysis ï All investors are risk-averse. ï The expected returns, variances, and covariances of all assets are known. ï To determine optimal portfolios, investors only need to know the expected returns, variances, and covariances of returns. ï There are no transactions costs or taxes.
Efficient Frontier ï A portfolio is efficient if the portfolio offers the highest level of expected return for a given level of volatility. ï The portfolios that have the greatest expected return for each level of volatility make up the efficient frontier.
The Effect of Correlation ï If ρæ, ̃ = 1, no diversification. The portfolio's volatility is simply the weighted average volatility of the two risky assets. ï If ρæ, ̃ < 1, the portfolio's volatility is reduced due to diversification. It is less than the weighted average volatility of the two risky assets. ï If ρæ, ̃ = −1, a zero-risk portfolio can be constructed.
Expected Return
Volatility
5% 0% 5% 10%
10%
15%
15%
20%
20%
25%
25%
30%
No Risk
Corr. = -
Corr. = +
BA
WMT
Number of Stocks
Systematic Risk
Portfolio Volatility
Elimination of Non-Systematic Risk
Portfolio Standard Deviations
Efficient Frontier
Expected Return
Calculating Beta
βæ = βæ, ̄îH = Cov[Ræ, R ̄îH
]
σ ̄îHà
= ρæ, ̄îH ⋅ σσæ ̄îH
βB = ⁄ xæβæ
ï
æ¡_ ρæ, ̄îH ⋅ σæ represents the systematic risk of i.
Beta can be estimated using linear regression: Ræ − rv = αæ + βæ(R ̄îH − rv) + εæ
Capital Asset Pricing Model (CAPM) ræ = E[Ræ] = rv + βæ[E[R ̄îH] − rv] ï E[R ̄îH] − rv is the market risk premium or the expected excess return of the market. ï E[Ræ] − rv or βæ[E[R ̄îH] − rv] is the risk premium for security i or the expected excess return of security i. Since βæ only captures systematic risk, E[Ræ] under the CAPM is not influenced by nonsystematic risk.
Assumptions of the CAPM ï Investors can buy and sell all securities at competitive market prices. There are no taxes or transaction costs. Investors can borrow and lend at the risk-free interest rate. ï Investors hold only efficient portfolios of traded securities. ï Investors have homogeneous expectations regarding the volatilities, correlations, and expected returns of securities. The consequence of these assumptions is that the market portfolio is the efficient portfolio.
Security Market Line (SML) SML is a graphical representation of CAPM:
CML vs. SML
CML SML The x-axis is based on total risk (i., volatility)
The x-axis is based on systematic risk (i., beta) Only holds for efficient portfolios (b/c all combinations of the risk-free asset and the market portfolio are efficient portfolios)
Holds for any security or combination of securities (b/c the CAPM can be used to calculate the expected return for any security)
Alpha The difference between a security's expected return and the required return (as predicted by the CAPM) is called alpha: αæ = E[Ræ] − ræ = E[Ræ] − (rv + βæ[E[R ̄îH] − rv]) If the market portfolio is efficient, then all securities are on the SML, and: E[Ræ] = ræ and αæ = 0 If the market portfolio is not efficient, then the securities will not all lie on the SML, and: E[Ræ] ≠ ræ and αæ ≠ 0 Investors can improve the market portfolio by: ï buying stocks whose E[Ræ] > ræ (i., αæ > 0) ï selling stocks whose E[Ræ] < ræ (i., αæ < 0)
Required Return on New Investment Adding the new investment will increase the Sharpe ratio of portfolio P if its expected return exceeds its required return, defined as: r‰} ̆ = rv + β‰} ̆,B[E[RB] − rv] In general, the beta of an asset i with respect to a portfolio P is: βæ,B = Cov[Ræ, RB
]
σBà
= ρæ,B ⋅ σ σæ B
Expected Returns and the Efficient Portfolio A portfolio is efficient if and only if the expected return of every available asset equals its required return. Thus, we have: E[Ræ] = ræ = rv + βæ}vv[E[R}vv] − rv]
Market Risk Premium Two methods to estimate the market risk premium: ï The historical risk premium: Uses the historical average excess return of the market over the risk-free interest rate. ï A fundamental approach: Uses the constant expected growth model to estimate the market portfolio’s expected return. P% =
Div_ E[R ̄îH] − g ⇒ E[R ̄îH
] = Div_ P% + g
The Debt Cost of Capital Two methods to estimate debt cost of capital: ï Adjustment from debt yield: rt = y − pL = Yield to mat. − Pr(Default) E[Loss rate] ï CAPM using debt betas: rt = rv + βt[E[R ̄îH] − rv] Note that: ï It is difficult to get the beta estimates for individual debt securities because they are traded less frequently than stocks. ï The average beta for debt tends to be low. However, the beta for debt does increase as the credit rating decreases.
Required Return on All-Equity Project If a project is financed purely with equity, the equity is said to be unlevered; otherwise, it is levered.
Assuming a project is financed entirely with equity, we can estimate the project’s cost of capital and beta based on the asset or unlevered cost of capital and the beta of comparable firms:
Comparable All- Equity
Levered
Beta βμ = β~
βμ
= wβ
- w ̧β ̧
Cost of Capital
rμ = r~
rμ
= wr
- w ̧r ̧
A firm's enterprise value is the risk of the firm's underlying business operations that is separate from its cash holdings. It is the combined market value of the firm's equity and debt, less any excess cash: V = E + D − C
To determine the enterprise value, we use the firm's net debt: Net debt = Debt − Excess cash and short-term investments
The beta of the firm's underlying business
enterprise is:
βμ = wβ + w ̧β ̧ + wŒβŒ
where:
w~ =
E
E + D − C
w ̧ =
D
E + D − C
wΠ=
−C
E + D − C
Required Return on a Leveraged Project
If the project is financed with both equity and debt, then use the weighted-average cost of capital (WACC):
r ̋{ŒŒ = wr + w ̧r ̧(1 − τŒ)
= rμ − w ̧r ̧τŒ
where r ̧(1 − τŒ) is the effective after-tax cost of debt.
Note that: ï WACC is based on the firm’s after-tax cost of debt while the unlevered cost of capital is based on the firm’s pretax cost of debt. ï Unlevered cost of capital is also called the asset cost of capital or the pretax WACC. ï When we say “WACC” with no qualification, we mean “after-tax WACC”.
Multi-Factor Model ï If the market portfolio is not efficient, a multi-factor model is an alternative. ï It considers more than one factor when estimating the expected return. ï An efficient portfolio can be constructed from other well-diversified portfolios. ï Also known as the Arbitrage Pricing Theory (APT). ï Similar to CAPM, but assumptions are not as restrictive.
Key Equations Using a collection of N factor portfolios:
E[Ræ] = rv + ⁄ βæúï(E[
‰
ï¡_
Rúï] − rv)
where: ï βæú_, ... , βæúï are the factor betas of asset i that measure the sensitivity of the asset to a particular factor, holding other factors constant. ï E[Rúï] − rv is the risk premium or the expected excess return for a factor portfolio.
If all factor portfolios are self-financing, then we can rewrite the equation as:
E[Ræ] = rv + ⁄ βæúï(E[
‰
ï¡_
Rúï])
For a self-financing portfolio, the portfolio weights sum to zero rather than one.
Fama-French-Carhart (FFC) This model consists of 4 self-financing factor portfolios: ï Market portfolio. Accounts for equity risk. Take a long position in the market portfolio and finance itself with a short position in the risk-free asset. ï Small-minus-big (SMB) portfolio. Accounts for differences in company size based on market capitalization. Buy small firms and finance itself by short selling big firms. ï High-minus-low (HML) portfolio. Accounts for differences in returns on value stocks and growth stocks. Buy high book-to-market stocks (i., value stocks) and finance itself by short selling low book-to-market stocks (i., growth stocks). ï Momentum. Accounts for the tendency of an asset return to be positively correlated with the asset return from the previous year. Buy the top 30% stocks and finance itself by short selling the bottom 30% stocks.
The FFC estimates the expected return as:
E[Ræ] = rv + βæ ̄îH(E[R ̄îH] − rv) + βæ≠ ̄ÔE[R≠ ̄Ô] + βæˇ ̄¥E[Rˇ ̄¥] + βæB!"!E[RB!"!] where: SMB = Small-minus-big portfolio HML = High-minus-low portfolio PR1YR = Prior 1-year momentum portfolio
Other Anomalies ï Siamese twins: Two stocks with claims to a common cash flow should be exposed to identical risks but perform differently. ï Political cycle effect: For a given political administration, its first year and last year yield higher returns than the years in between. ï Stock split effect: Returns are higher before and after the company announces the stock split. ï Neglected firm effect: Lesser-known firms yield abnormally high returns. ï Super Bowl effect: Historical data shows in the year after the Super Bowl, the stock market is more likely to do better if an NFC team won and worse if an AFC team won. ï Size effect: Small-cap companies have outperformed large-cap companies on a risk-adjusted basis. ï Value effect: Value stocks have consistently outperformed growth stocks.
Bubbles also violate market efficiency. It happens when the market value of the asset significantly deviates from its intrinsic value.
The Efficiency of the Market Portfolio Under the CAPM assumptions, the market portfolio is an efficient portfolio. All investors should hold the market portfolio (combined with risk-free investments). This investment advice does not depend on the quality of an investor’s information or trading skill.
The assumption of rational expectations is less rigid than that of homogeneous expectations. If we assume investors have rational expectations, then all investors correctly interpret and use their own information, along with information from market prices and the trades of others.
The market portfolio can be inefficient (and thus it is possible to beat the market) only if a significant number of investors: ï Do not have rational expectations (thus information is misinterpreted). ï Care about aspects of their portfolio other than expected return and volatility (thus they are willing to hold portfolios that are mean-variance inefficient).
Takeover Offer: ï After the initial jump in the stock price at the time of the announcement, target stocks do not appear to generate abnormal subsequent returns on average. ï Stocks that are ultimately acquired tend to appreciate and have positive alphas, while stocks that are not acquired tend to depress and have negative alphas.
Stock Recommendation: ï When a stock recommendation is given at the same time that news about the stock is released, the initial stock price reaction appears correct. The stock price increases in the beginning, then it flattens out. ï When a stock recommendation is given without news, the stock price seems to overreact. The stock price surges the following day, then it falls compared to the market
The Performance of Fund Managers: ï The median mutual fund actually destroys value. ï The mutual fund industry still has positive value added because skilled managers manage more money and add value to the whole industry. ï On average, an investor does not profit more from investing in an actively managed mutual fund compared to investing in passive index funds. The value added by a fund manager is offset by the mutual fund fees. ï Superior past performance of funds was not a good predictor of future ability to outperform the market
Reasons Why the Market Portfolio Might Not Be Efficient: ï Proxy Error: Due to the lack of competitive price data, the market proxy cannot include most of the tradable assets in the economy. ï Behavioral Biases: Investors may be subject to systematic behavioral biases and therefore hold inefficient portfolios. ï Alternative Risk Preferences: Some investors focus on risk characteristics other than the volatility of their portfolio, and they may choose inefficient portfolios as a result. ï Non-Tradable Wealth: Investors are exposed to significant risks outside their portfolio. They may choose to invest less in their respective sectors to offset the inherent exposures from their human capital.
The Behavior of Individual Investors The following behaviors do not impact the efficiency of the market and have no effect on market prices or returns.
Underdiversification Individual investors fail to diversify their portfolios adequately. They invest in stocks of companies that are in the same industry or are geographically close. Explanations: ï Investors suffer from familiarity bias, favoring investments in companies they are familiar with. ï Investors have relative wealth concerns, caring most about how their portfolio performs relative to their peers.
Excessive Trading and Overconfidence Individual investors tend to trade very actively. Explanations: ï Overconfidence bias. They often overestimate their knowledge or expertise. Men tend to be more overconfident than women. ï Trading activity increases with the number of speeding tickets an individual receives – sensation seeking.
Systematic Trading Biases The following behaviors lead investors to depart from the CAPM in systematic ways and subsequently impact the efficiency of the market.
Holding on to Losers and the Disposition Effect Investors tend to hold on to investments that have lost value and sell investments that have increased in value. ï People tend to prefer avoiding losses more than achieving gains. They refuse to “admit a mistake” by taking the loss. ï Investors are more willing to take on risk in the face of possible losses. ï The disposition effect has negative tax consequences.
Investor Attention, Mood, and Experience ï Individual investors tend to be influenced by attention-grabbing news or events. They buy stocks that have recently been in the news. ï Sunshine has a positive effect on mood and stock returns tend to be higher on a sunny day at the stock exchange. ï Major sports events have impacts on mood. A loss in the World Cup reduces the next day’s stock returns in the losing country. ï Investors appear to put inordinate weight on their experience compared to empirical evidence. People who grew up during a time of high stock returns are more likely to invest in stocks.
Herd Behavior Investors actively try to follow each other's behavior. Explanations: ï Investors believe others have superior information, resulting in information cascade effect. ï Investors follow others to avoid the risk of underperforming compared to their peers (relative wealth concerns). ï Investment managers may risk damaging their reputations if their actions are far different from their peers. If they feel they are going to fail, then they would rather fail with most of their peers than fail while most succeed.
INVESTMENT RISK AND PROJECT
ANALYSIS
Measures of Investment Risk Variance ï The average of the squared deviations above and below the mean: Variance = E[(R − E[R])à]
Semi-variance / Downside Semi-variance ï Only cares about downside risk; ignores upside variability. ï The average of the squared deviations below the mean: Semi-variance = E[min(0, R − E[R])à] ï The sample semi-variance is: Semi-variance = 1n ∑ min(0, Ræ − E[R])à ï Semi-variance ≤ Variance ï For a symmetric distribution: Semi-variance = 1 2 Variance
Value-at-Risk (VaR) ï VaR of X at the 100α% confidence level is its 100αth percentile, denoted as VaR±(X) or π±. Pr[X ≤ π±] = α ï If X is continuous, then: F†(π±) = α ⇒ π± = F†O_(α) ï VaR%.¢%(X) is the 50th percentile or the median of X.
Tail Value-at-Risk (TVaR) ï TVaR focuses on what happens in the adverse tail of the probability distribution. ï Also known as the conditional tail expectation or expected shortfall. ï If X represents gains, then the risk we are concerned about comes from the low end of the distribution: TVaR± = E[X|X ≤ π±] = 1α fl x ⋅ f†(x)dx
$% O· ï If X represents losses, then the risk we are concerned about comes from the high end of the distribution: TVaR± = E[X|X > π±]
=
1
1 − α fl x ⋅ f†
(x)dx
· $%
ï If the risk we are concerned about is unclear, then use the following rule of thumb: - If α < 0, then presumably the risk of concern comes from the low end. - If α > 0, then presumably the risk of concern comes from the high end. ï TVaR will provide a more conservative number than VaR.
Coherent Risk Measures g(X) is coherent if it satisfies (for c > 0): ï Translation invariance: g(X + c) = g(X) + c ï Positive homogeneity: g(cX) = c ⋅ g(X) ï Subadditivity: g(X + Y) ≤ g(X) + g(Y) ï Monotonicity: If X ≤ Y, then g(X) ≤ g(Y)
Variance, Semi-Variance, VaR, TVaR: ï Variance and semi-variance do not satisfy any of the 4 characteristics; not coherent. ï VaR is usually not coherent since it does not satisfy the subadditivity characteristic. If the distributions are assumed to be normal, then VaR can be shown to be coherent. ï TVaR is always coherent.
Project Risk Analysis The net present value (NPV) of a project equals the present value of all expected net cash flows from the project. The discount rate for a project is its cost of capital.
Breakeven Analysis ï Calculate the value of each parameter so that the project has an NPV of zero. ï The internal rate of return (IRR) is the rate at which the NPV is zero.
Sensitivity Analysis ï Change the input variables one at a time to see how sensitive NPV is to each variable. Using this analysis, we can identify the most significant variables by their effect on the NPV. ï The range is the difference between the best-case NPV and the worst-case NPV.
INVESTMENT RISK AND
PROJECT ANALYSIS
Venture Capital Financing Terms Venture capitalists typically hold convertible preferred stock, which differs from common stock due to: ï Liquidity preference Liquidity preference = Multiplier × Initial inv ï Participation rights ï Seniority ï Anti-dilution protection ï Board membership
There are two ways to exit from a private company: ï Acquisition ï Public offering
Initial Public Offering An initial public offering (IPO) is the first time a company sells its stock to the public.
Advantages of IPO: ï Greater liquidity ï Better access to capital
Disadvantages of IPO: ï Dispersed equity holdings ï Compliance is costly and time-consuming
There are two major types of offerings: ï Primary offerings: New shares sold to raise new capital. ï Secondary offerings: Existing shares sold by current shareholders.
When issuing an IPO, the company and underwriter must decide on the best mechanism to sell shares: ï Best-efforts: Shares will be sold at the best possible price. Usually used in smaller IPOs. ï Firm commitment: All shares are guaranteed to be sold at the offer price. Most common. ï Auction IPOs: Shares sold through an auction system and directly to the public.
Standard steps to launching a typical IPO:
- Underwriters typically manage an IPO and they are important because they: o Market the IPO. o Assist in required filings. o Ensure the stock’s liquidity after the IPO.
- Companies must file a registration statement, which contains two main parts: o Preliminary prospectus/red herring. o Final prospectus.
- A fair valuation of the company is performed by the underwriter through road show and book building.
- The company will pay the IPO underwriters an underwriting spread. After the IPO, underwriters can protect themselves more against losses by using the over-allotment allocation or greenshoe provision.
4 IPO Puzzles: ï The average IPO seems to be priced too low. ï New issues appear cyclical. ï The transaction costs of an IPO are high. ï Long-run performance after an IPO is poor on average.
Debt Financing Corporate Debt: Public Debt Public debt trades on public exchanges. The bond agreement takes the form of an indenture, which is a legal agreement between the bond issuer and a trust company. 4 common types of corporate debt: ï Notes (Unsecured) ï Debentures (Unsecured) ï Mortgage bonds (Secured) ï Asset-backed bonds (Secured)
The new debt that has lower seniority than existing debenture issues is called a subordinated debenture.
International bonds are classified into four broadly defined categories: ï Domestic bonds – issued by local, bought by foreign ï Foreign bonds – issued by foreign, bought by local ï Eurobonds – issued by local or foreign ï Global bonds
Corporate Debt: Private Debt Private debt is negotiated directly with a bank or a small group of investors. It is cheaper to issue due to the absence of the cost of registration.
2 main types of private debt: ï Term loan ï Private placement
Other Types of Debt Government entities issue sovereign debt and municipal bonds to finance their activities.
Sovereign debt is issued by the national government. In the US, sovereign debt is issued as bonds called "Treasury securities."
There are four types of Treasury securities: ï Treasury bills ï Treasury notes ï Treasury bonds ï Treasury inflation-protected securities (TIPS)
Municipal bond is issued by the state and local governments.
There are also several types of municipal bonds based on the source of funds that back them: ï Revenue bonds ï General obligation bonds
Asset-Backed Securities An asset-backed security (ABS) is a security whose cash flows are backed by the cash flows of its underlying securities.
The biggest sector of the ABS market is the mortgage-backed security (MBS) sector. An MBS has its cash flows backed by home mortgages. Because mortgages can be repaid early, the holders of an MBS face prepayment risk.
Banks also issue ABS using consumer loans, such as credit card receivables and automobile loans.
A private ABS can be backed by another ABS. This new ABS is known as a collateralized debt obligation (CDO).
Capital Structure Theory: Perfect Capital Markets Perfect Capital Markets ï Investors and firms can trade the same set of securities at competitive market prices equal to the present value of their future cash flows. ï No taxes, transaction costs, or issuance costs. ï The financing and investment decisions are independent of each other.
MM Proposition I ï The total value of a firm is equal to the market value of the total cash flows generated by its asset. ï The value of a firm is unaffected by its choice of capital structure. ï Changing a firm's capital structure merely changes how the value of its assets is divided between debt and equity, but not the firm's total value. ï V¥ = Vμ
Homemade leverage: ï Investors can borrow or lend at no cost on their own to achieve a capital structure different from what the firm has chosen. ï If an investor adds $x worth of debt to the capital structure, then he must reduce the equity by $x in order for the total firm’s value to remain unchanged. To determine x, set the adjusted current debt-equity ratio to equal the target debt-equity ratio: D + x E − x = çDEé"ËA'}H
MM Proposition II ï The cost of capital of levered equity increases with the firm's debt-to-equity ratio: r~ = rμ + DE (rμ − r ̧)
ï Because there are no taxes in a perfect capital market, the firm’s WACC and the unlevered cost of capital coincide: rμ = r ̋{ŒŒ =
E
E + D r~ +
D
E + D r ̧
Note: ï Since debt holders have a priority claim on assets and income above equity holders, debt is less risky than equity, and thus r ̧ < r~. ï As companies take on more debt, the risk to equity holders increases, and subsequently the cost of equity increases. ï As the amount of debt increases, the chance that the firm will default increases, and subsequently the cost of debt increases. ï Although both cost of debt and cost of equity increase as the company takes on more debt, WACC remains unchanged because more weight is placed on the lower-cost debt.
WACC with Multiple Securities: rμ = r ̋{ŒŒ = ⁄ wæ ⋅ ræ
Levered and Unlevered Betas:
βμ = wβ + w ̧β ̧
β~ = βμ + DE (βμ − β ̧)
Capital Structure Theory: Taxes and Financial Distress Costs Interest Tax Shield ï The use of debt results in tax savings for the firm, which adds to the value of the firm. ï V¥ = Vμ + PV(Interest tax shield) Interest tax shield = Corp. Tax Rate × Int Pmt
For a firm that borrows debt D and keeps the debt permanently, if the firm's marginal tax rate (a.k. effective tax advantage of debt) is τŒ, then the present value of the interest tax shield is: PV(Interest tax shield) = τŒ ⋅ D
WACC with Taxes
The firm’s effective after-tax WACC
measures the required return to the firm’s
investors after taking into account the
benefit of the interest tax shield:
r ̋{ŒŒ = wr + w ̧r ̧(1 − τŒ)
= wr + w ̧r ̧ − w ̧r ̧τŒ
where:
wr + w ̧r ̧ = rμ = pretax WACC
w ̧r ̧τŒ = reduction due to tax shield
Note: ï As debt increases, the reduction due to interest tax shield increases, WACC falls, and thus the value of the firm increases.
Interest Tax Shield with a Target Debt- Equity Ratio When a firm adjusts its debt over time so that its debt-equity ratio is expected to remain constant, we can value the interest tax shield by:
- Calculating the value of the unlevered firm, Vμ, by discounting cash flows at the unlevered cost of capital (i., pre-tax WACC).
- Calculating the value of the levered firm, V¥, by discounting cash flows at the WACC (i., after-tax WACC).
- PV(Interest tax shield) = V¥ − Vμ
Financial Distress Costs A firm that fails to make its required payments to debt holders is said to default on its debt.
After the firm defaults, the debt holders have claims to the firm's assets through a legal process called bankruptcy.
Two forms of bankruptcies: ï Chapter 7 liquidation. A trustee supervises the liquidation of the firm's assets through an auction. The proceeds from the liquidation are used to pay the firm's creditors, and the firm ceases to exist. ï Chapter 11 reorganization. The firm's existing management is given the opportunity to propose a reorganization plan. While developing the plan, management continues to operate the business.
Ifm formula sheet
University: Iowa State University
